In all cases in which we treat the measure classification problem, i.e.  all cases except Case 2.3.b, it will be possible to address the equidistribution problem. In fact, Case 1 (trivial base) is precisely the setting of Benoist–Quint [1]} so the corresponding equidistribution results (see [2]}, [3]} extending the original results with respect to moment hypotheses) directly apply; we do not comment on it further here. Case 2.1 boils down to the equidistribution to the Furstenberg measure; even quantitative statements are known for this case, see §REF . Finally, thanks to the decomposability obtained in Theorem REF , it is not hard to see that Case 2.3.a also boils down to the setting of Benoist–Quint, see Proposition REF .
